# Countably Infinite Multilevel Source Polarization for Non-Stationary Erasure Distributions

## TL;DR

This paper extends the analysis of polar transforms to non-stationary sources over infinite alphabets, introducing a novel multilevel polarization framework based on group structures and elementary lattice theory.

## Contribution

It is the first to analyze source polarization over infinite alphabets, specifically for sources modeled as Polish groups, and provides recursive formulas for erasure distributions.

## Key findings

- First source polarization analysis over infinite alphabets.
- Recursive formulas for erasure distributions on Polish groups.
- Concrete examples of multilevel polarization with countably infinite levels.

## Abstract

Polar transforms are central operations in the study of polar codes. This paper examines polar transforms for non-stationary memoryless sources on possibly infinite source alphabets. This is the first attempt of source polarization analysis over infinite alphabets. The source alphabet is defined to be a Polish group, and we handle the Ar{\i}kan-style two-by-two polar transform based on the group. Defining erasure distributions based on the normal subgroup structure, we give recursive formulas of the polar transform for our proposed erasure distributions. As a result, the recursive formulas lead to concrete examples of multilevel source polarization with countably infinite levels when the group is locally cyclic. We derive this result via elementary techniques in lattice theory.